Partially averaged Navier-Stokes equations

Flows containing a transition to turbulence are inherent in a wide range of phenomena and processes, such as supernova explosions, combustion of gas mixtures, flow around bodies of various shapes, etc. Numerical simulation of such flows is of significant practical interest and is a complex independent task. To solve this problem, there are several main approaches in computational fluid dynamics, which have their own area of applicability, advantages and disadvantages. Thus, direct numerical simulation (DNS) and the large eddy simulation method (LES/ILES) are optimal approaches for describing flows in which there is a transition to turbulence, since they resolve a wide range of flow scales, but at the same time require significant computational resources due to the use of fine grids. Approaches based on the Reynolds Averaged Navier-Stokes equations (RANS) use a completely stochastic description of turbulence and have a significantly lower computational cost. At the same time, they allow one to describe only steady or slowly changing flows. A possible alternative is hybrid methods that combine the strengths of DNS/LES and RANS.
In this paper, it is considered a hybrid approach based on partially averaged Navier-Stokes equations (PANS), which provides a seamless transition from RANS to DNS/LES. A detailed derivation of the corresponding system of equations and theoretical estimates characterizing the possibilities of the approach are given.